S: Any hints on how to jump off from 8? It's the only one I'm still stuck on.
T: There are three types of similar problems, and you solve them all the same way:
a) highest power of x in the numerator and the denominator are equal 6x^2/3x^2
b) highest power of that of the denominator
c) highest power of x in the numerator is less than that of the denominator
Divide each term by the highest power of x in the denominator, then analyze what remains. Problem #8 is an example of type c. When you divide through by x^5, remember that 1/x approaches 0 when x approaches infinity.
Let me know if you want me to continue, OK?
I love Donna's response to the student's question. She gives him more information about similar problems but never tells him specifically how to solve it. Studies have shown that American teachers don't do enough of this. We tend to show them how to solve it without giving them the opportunity, with a little guidance, to solve it themselves. This example shows one reason why her students are doing so well in this course and will probably do very well on the AP exam.
Student-Student interaction
The students were asked to introduce themselves from their pet's perspective:
S1: Hey, I'm Hudson, a Nova Scotia Duck Tolling Retriever, and I'd be happy to tell you a bit about one of my family members, Nikolai. On weekdays, he usually get home from his soccer, skiing, or tennis practices at around 6, and flops around on the couch until he's fed. After dinner, he practices cello (what a horrible sound!) and does his homework, usually right up until bedtime. But, if I'm lucky, Nikolai will throw my favorite ball around before we both go to sleep. On the weekends, he'll often take me walking down by the river in the morning so I can go swimming, or so he can go swimming with his friends. Other than that, I usually don't see him much on weekends as he is usually out playing cello somewhere or with his friends.
S2: I don't play the cello which I think is cool but I do play the guitar and viola. I like the viola best. Do you play any other instruments?
S3: Cool, you are a violist? Not a common instrument, at least in my school and my surroundings! I play the violin, and whenever I have the chance to try the viola, I like the wider and thicker feeling of the fingerings!
S2: Ha ha ha yeah! I've been playing since 5th grade. I love it. I tried the violin for a while but I just didn't like the E string. I like the lower C string. I also like playing Irish/Celtic songs, which allow my fingers and bow to move at a fast pace. J
I chose this artifact because it shows how Donna often interacts with her students. They usually start with a question, and she answers with more information but not the solution. Studies have shown that American mathematics teachers do not do enough of this. We tend to give our students the answers or solutions as soon as they beign to struggle. Those same studies give this as a possible reason for our students performing so far below students of other countries in international contests. Donna acts as a guide throughout this course. She shares the new content with them and then guides them through the applications. It is now obvious why her students are so successful in this course and probably will be equally successful on the AP Exam.
S: Any hints on how to jump off from 8? It's the only one I'm still stuck on.
T: There are three types of similar problems, and you solve them all the same way:
a) highest power of x in the numerator and the denominator are equal 6x^2/3x^2
b) highest power of that of the denominator
c) highest power of x in the numerator is less than that of the denominator
Divide each term by the highest power of x in the denominator, then analyze what remains. Problem #8 is an example of type c. When you divide through by x^5, remember that 1/x approaches 0 when x approaches infinity.
Let me know if you want me to continue, OK?
I love Donna's response to the student's question. She gives him more information about similar problems but never tells him specifically how to solve it. Studies have shown that American teachers don't do enough of this. We tend to show them how to solve it without giving them the opportunity, with a little guidance, to solve it themselves. This example shows one reason why her students are doing so well in this course and will probably do very well on the AP exam.
- Student-Student interaction
The students were asked to introduce themselves from their pet's perspective:S1: Hey, I'm Hudson, a Nova Scotia Duck Tolling Retriever, and I'd be happy to tell you a bit about one of my family members, Nikolai. On weekdays, he usually get home from his soccer, skiing, or tennis practices at around 6, and flops around on the couch until he's fed. After dinner, he practices cello (what a horrible sound!) and does his homework, usually right up until bedtime. But, if I'm lucky, Nikolai will throw my favorite ball around before we both go to sleep. On the weekends, he'll often take me walking down by the river in the morning so I can go swimming, or so he can go swimming with his friends. Other than that, I usually don't see him much on weekends as he is usually out playing cello somewhere or with his friends.
S2: I don't play the cello which I think is cool but I do play the guitar and viola. I like the viola best. Do you play any other instruments?
S3: Cool, you are a violist? Not a common instrument, at least in my school and my surroundings! I play the violin, and whenever I have the chance to try the viola, I like the wider and thicker feeling of the fingerings!
S2: Ha ha ha yeah! I've been playing since 5th grade. I love it. I tried the violin for a while but I just didn't like the E string. I like the lower C string. I also like playing Irish/Celtic songs, which allow my fingers and bow to move at a fast pace. J
I chose this artifact because it shows how Donna often interacts with her students. They usually start with a question, and she answers with more information but not the solution. Studies have shown that American mathematics teachers do not do enough of this. We tend to give our students the answers or solutions as soon as they beign to struggle. Those same studies give this as a possible reason for our students performing so far below students of other countries in international contests. Donna acts as a guide throughout this course. She shares the new content with them and then guides them through the applications. It is now obvious why her students are so successful in this course and probably will be equally successful on the AP Exam.